Question

Graph the function f(x)=
–
2(x+6)2.

Answer 1

To graph the function \( f(x) = -2(x+6)^2 \), we can break it down into key steps:

1. **Identify the Type of Function**: This function is a quadratic function in vertex form \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( a = -2 \), \( h = -6 \), and \( k = 0 \).

2. **Determine the Vertex**: From the vertex form, we can easily determine the vertex:
\[
\text{Vertex} = (-6, 0)
\]

3. **Determine the Direction of the Parabola**: Since \( a = -2 \) is negative, the parabola opens downward.

4. **Find the Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex:
\[
x = -6
\]

5. **Calculate Intercepts**:
- **Y-Intercept**: Set \( x = 0 \):
\[
f(0) = -2(0 + 6)^2 = -2(6^2) = -2(36) = -72
\]
So the y-intercept is \( (0, -72) \).

- **X-Intercepts**: Set \( f(x) = 0 \):
\[
-2(x + 6)^2 = 0 \implies (x + 6)^2 = 0 \implies x + 6 = 0 \implies x = -6
\]
Thus, the x-intercept is \( (-6, 0) \).

6. **Plot Additional Points**: Choose some x-values around the vertex and calculate corresponding f(x) values:
- For \( x = -7 \):
\[
f(-7) = -2(-7 + 6)^2 = -2(-1)^2 = -2
\]
- For \( x = -5 \):
\[
f(-5) = -2(-5 + 6)^2 = -2(1)^2 = -2
\]

7. **Sketch the Graph**:
- Plot the vertex \( (-6, 0) \).
- Plot the y-intercept \( (0, -72) \).
- Plot the x-intercept \( (-6, 0) \).
- Plot additional points \( (-7, -2) \) and \( (-5, -2) \).
- Draw a downward-opening parabola through these points.

The graph should reflect a downward-opening parabola with its vertex at \( (-6, 0) \), symmetrical around the axis of symmetry \( x = -6 \).

Now, you can sketch the graph using the points found, ensuring it opens downward and passes through the intercepts.